A small dynamic leaf-level model predicting photosynthesis in greenhouse tomatoes

The conversion of supplemental greenhouse light energy into biomass is not always optimal. Recent trends in global energy prices and discussions on climate change highlight the need to reduce our energy footprint associated with the use of supplemental light in greenhouse crop production. This can be achieved by implementing “smart” lighting regimens which in turn rely on a good understanding of how fluctuating light influences photosynthetic physiology. Here, a simple fit-for-purpose dynamic model is presented. It accurately predicts net leaf photosynthesis under natural fluctuating light. It comprises two ordinary differential equations predicting: 1) the total stomatal conductance to CO2 diffusion and 2) the CO2 concentration inside a leaf. It contains elements of the Farquhar-von Caemmerer-Berry model and the successful incorporation of this model suggests that for tomato (Solanum lycopersicum L.), it is sufficient to assume that Rubisco remains activated despite rapid fluctuations in irradiance. Furthermore, predictions of the net photosynthetic rate under both 400ppm and enriched 800ppm ambient CO2 concentrations indicate a strong correlation between the dynamic rate of photosynthesis and the rate of electron transport. Finally, we are able to indicate whether dynamic photosynthesis is Rubisco or electron transport rate limited.


Introduction
The cultivation of greenhouse crops under optimised conditions will become increasingly important, with the need for year-round crop harvesting under changing environmental conditions as a driving factor. The widespread use of supplemental lighting to optimise growing conditions is a key tool at our disposal. The efficiency of converting light energy into plant photosynthesis is however not always optimal. This is particularly true when lights are first turned on, where time is required for the activation of key photosynthetic enzymes and for adjustments in stomatal pore aperture [1].
Given recent trends in global energy prices and continuous discussions on climate change, our energy footprint associated with the use of supplemental light needs to be reduced. Two potential avenues that may lead to the efficient conversion of light energy into photosynthesis are: 1) the use of light-emitting diodes (LEDs) in supplemental lighting applications. LEDs increase the efficiency with which electrical energy is converted to photosynthetically active radiation (PAR), the radiation with wavelength 400-700nm, which powers photosynthesis [2]. 2) A model-based implementation of "smart" lighting regimens. This approach necessitates a good understanding of how plants harness light energy under natural fluctuating irradiance (I).
Plant responses to fluctuating irradiance occur across numerous levels of complexity, ranging from the whole canopy (at crop level and governed by plant structure) to the biochemistry of a single reaction (at leaf level and governed by physiology) and across different orders of magnitude in time. At the top of the canopy light levels depend on factors such as the solar angle and amount of cloud cover. Lower parts of a crop canopy rely on light in the form of sun-flecks, and the amount of these in turn depends on the canopy's structure [3].
Light intensity is the most dynamic condition to which greenhouse crops need to respond and can change at time scales ranging from a season (winter versus summer) to less than 1 second (passing clouds) [4]. The variation of incident light on leaves in the upper canopy can have a substantial effect on photosynthesis in this upper layer since it accounts for up to 75% of crop canopy carbon assimilation [5].
At leaf level, a plant's ability to regulate photosynthesis in response to rapid variations in irradiance may be restricted by the following factors: 1) the opening/closing of stomata, 2) the activation/deactivation of Calvin cycle enzymes, 3) the up-regulation/down-regulation of photoprotective processes [3], 4) transiently changing mesophyllic conductance [6].
Previous leaf-level models have ranged both in complexity and in time scales of prediction. Depending on the research question, a model may for example include detailed mechanisms such as the regulation of enzymatic activity and both photochemical and non-photochemical quenching. The result is a model that comprises a multitude of differential and rate equations, which may include time constants in different scales (as an example consider the e-photosynthesis model published in 2013 [7] with 75 differential equations and approximately 120 rate equations).
These models often include the well-know Farquhar-von Caemmerer-Berry (FvCB) model, which mathematically describes key Calvin cycle processes and linear transport rates [8]. The result is an estimated net photosynthetic rate (A n ) which stems from competitive enzymatic processes involving CO 2 and O 2 binding under different environmental conditions [9]. A brief summary of selected small dynamic models is given in Table 1. We included a summary of the model presented here for comparison.
The rest of this article is organised as follows. In section 1 we provide the theoretical background to the model and describe the plant physiology underpinning the well-known FvCB model. We also discuss three of the factors that may restrict A n in greater detail. The model is defined in section 2, and the materials and methods used are discussed in section 3. Results and discussion are presented in sections 4 and 5, respectively, and conclusions are given in section 6.

Theory
We briefly introduce three of the factors that affect A n , and which are included in the model: 1) stomatal conductance, 2) the Rubisco limited carboxylation rate, and 3) the electron transport limited carboxylation rate.
They do so by adjusting their pore aperture and this is achieved by changing the form of their 2 guard cells. These structures can, therefore, be thought of as conductors of CO 2 diffusion and, therefore, only allow a certain amount of CO 2 to enter a leaf.
Stomatal aperture depends on numerous environmental cues. For C3 plants (plants that allow for the direct carbon fixation of CO 2 ) under non-limiting growth conditions, pore sizes increase with increased irradiance and low CO 2 concentrations [13].
The metabolic regulation of these guard cells is highly complex and accordingly, an empirical model that predicts dynamic changes in the conductance of CO 2 related to perturbations in irradiance is used here [12]. Refer to S1 File for a discussion on how the total stomatal conductance to CO 2 diffusion (g tc ) is defined.

The FvCB model
This model describes the steady-state net photosynthetic (A n ) rate as [8,14]: where 1 defines A n as the difference between the gross photosynthetic rate (A g ) and R d , the mitochondrial respiration which includes the release of CO 2 in light other than photo-respiration [14]. From 2 follows that A n can be limited by the Rubisco limited carboxylation rate (W c ), the electron transport limited carboxylation rate (W j ) or the rate at which triose phosphates are utilised (W p ). Opting to keep our model structure concise and the number of unknown system parameters to a minimum, we omit the limiting factor W p . The Rubisco limited carboxylation rate (W c ). Once CO 2 enters a leaf through the stomata, it diffuses through the inter-cellular spaces into the chloroplasts by means of a diffusion gradient between the chloroplast and the rest of the leaf. The numerous photosynthesis reactions that occur inside the chloroplast are summarised in the Calvin cycle, a process which comprises both light dependent and independent reactions.
During the first phase of this cycle, a single CO 2 molecule is fixed onto a Ribulose 1,5-bisphosphate (RuBP) molecule to form two 3-Phosphoglyceric acid (3-PGA) molecules. Important here in the context of modelling enzyme kinetics is that this process is catalysed by the enzyme Ribulose-1,5-bisphosphate carboxylase/oxygenase (Rubisco), and its activation state is in turn increased by Rubisco activase. The structure of 3-PGA allows it to be combined and rearranged to form sugars which can be transported or stored for energy. The rate at which CO 2 fixation takes place is known as the carboxylation rate. However, Rubisco also catalyses RuBP oxygenation (binding RuBP to O 2 ). This reduces the efficiency of the Calvin cycle. The rate at which this takes place is called the oxygenation rate.
Mathematically, the Rubisco limited carboxylation rate is given as [8,14,15]: where V cmax is the maximum obtainable carboxylation rate, and c c is the partial pressure of CO 2 in the chloroplast stroma whereas O c is the partial pressure of O 2 in the chloroplast stroma. K c is the Michealis-Menten constant for CO 2 , K o is the Michaelis-Menten constant for O 2 , [R] is the concentration of unbound (available) RuBP, and K r ' is the effective Michealis-Menten constant for RuBP. Assuming that RuBP is in excess, (3) reduces to [16], A relationship between W c and the oxygenation rate is introduced in 2 by Γ � , the CO 2 concentration at which oxygenation proceeds at twice the rate of carboxylation causing the photosynthetic uptake of CO 2 to be compensated for by the photorespiratory release of CO 2 [17].
Notice that expression 4 is defined for CO 2 concentrations in the chloroplast stroma (c c ). This concentration can however not be measured directly and so is predicted if A n , c i , and g m are known, by, where g m is the mesophyllic conductance encountered along the CO 2 diffusion pathway. By assuming that this conductance is infinitely large 5 simplifies to c c = c i . Electron transport limited carboxylation rate (W j ). The synthesis of RuBP also requires energy in the form of ATP and NADPH, and both ADP and NADP + are continuously converted to these energy supplying molecules in light dependent reactions which are dependent on the rate of electron transport (J). Accordingly, the electron transport limited carboxylation rate (W j ) is given as [8,14], Eq 6 assumes 4 electrons per carboxylation and oxygenation and so, based on the number of electrons required for NADP+ reduction, the standard values used are 4 and 8. However, there are uncertainties in the relationship between electron transport and ATP synthesis. For example, 4.5 and 10.5 have also been used [17]. Given the above mentioned assumptions, expression 2 can be written as

Temperature and light intensity effects on steady-state photosynthesis
The dependence of key FvCB model parameters on both leaf temperature (T l ) and irradiance (I) was already included in the 1980 publication by Farquhar et al. [8].
Changes in the values of parameters Γ � , K o , and K c (in 7) associated with leaf temperature changes are described using the Arrhenius equation [17], Here, the ideal gas constant R 1 , is expressed in units kJmol −1 K −1 . Constants ΔH ai and c i , where i = 1, . . ., 3, are the respective energies of activation and scaling constants for parameters Γ � , K o and K c .
Electron transport becomes limited when insufficient quanta are absorbed [8]. Accordingly, J is modelled as a function of irradiance [18], ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi where parameter J max is the upper limit to potential chloroplast electron transport determined by the components of the chloroplast electron transport chain [14]. Parameters θ and γ are unit-less (refer to S1 File for details). All parameter values are given in Table 3.

Dynamic model structure
The model we present in this paper comprises only two ordinary differential equations, the first predicting the total stomatal conductance to CO 2 diffusion (g tc ) and the second the CO 2 concentration inside the leaf (c i ). Here, g tc is the sum of the boundary layer and stomatal conductances (see S1 File). An algebraic relation between the predicted states is used to approximate the net photosynthetic rate (A n ). An overview of the dynamic states, system parameters, model inputs, and the measured output is given by the standard state-space representation, Function f is defined in Eqs 15-18. States g tc and c i , and the predicted A n are measured model outputs. The three system parameters that need to be inferred from the measured data are k u , k d and c 3 in expressions 10, and 15 and 16, respectively. Three environmental conditions, I(t), c a (t) and T l (t), are directly measurable and modelled as inputs/disturbances to the system. Predictions for A n are made using Fick's law of diffusion. Also known as the net flux of CO 2 that enters a leaf, the dynamic A n (achieved at a specific light intensity, as opposed to the attainable steady-state values predicted in 7) is calculated as [19,20] A n ðtÞ ¼ g tc ðtÞðc a ðtÞ À c i ðtÞÞ ð14Þ The asymmetric exponential response of stomata to increases and decreases in irradiance has often been reported [21]. This is modelled by introducing 2 time constants to the system, k u describing the rate of increase in g tc observed with an increase in irradiance, and k d describing the rate of decrease in g tc following a decrease in irradiance [10,13,[21][22][23][24][25]. The resulting model structure is, G(I(t),c a (t)) can be interpreted as the steady-state target function of g tc for a particular combination of I(t) and c a (t). A description of how G(I(t),c c (t)) should be calibrated is given in S1 File. The CO 2 concentration inside the leaf is modelled using a mass balance equation (refer to S1 File for a discussion), The minimum function stems from the FvCB model in Eq 7. Substituting the functions W c (t) and W j (t), that have been adjusted to take T l and I into account, into 17 gives,

Growing conditions of plants
Tomato plants were cultivated in a climate chamber (size: 16 m 2 ) in Wageningen University &amp; Research, Wageningen, the Netherlands (52˚N, 6˚E). Seeds were germinated in rockwool plugs (diameter: 2 cm) and transferred to rockwool cubes (10 × 10 × 7 cm) one week after sowing. Unless stated otherwise, day and night temperatures were set at 23˚C and 20˚C, respectively. Relative humidity was set at 70%. The CO 2 concentration was kept at ambient (450 ppm). Plants were irrigated automatically twice per day using an ebb & flow system (at 7 AM and 7 PM) with tomato nutrient solution (EC:2.2±0.1 mScm −1 , pH:5.5) (see S1 File).
Plants used to estimate V cmax and R d . These plants were exposed to an irradiance of 250 μmolm −2 s −1 provided by two types of high-pressure sodium lamps (SON-T and HPI-T PLUS, Philips Lighting). The photoperiod in the chamber was 16 hours. The SON-T lamps were switched on one hour before the HPI-T PLUS lamps and were switched off one hour after the HPI-T PLUS lamps in an attempt to mimic the gradual increase and decrease of irradiance during sunrise and sunset.
Plants used to parameterise function J(I(t)). The plants were exposed to an average irradiance of 200 μmolm −2 s −1 , with irradiance fluctuating between 50 and 500 μmolm −2 s −1 every minute. The photoperiod in the chamber was 16 hours. The irradiance pattern was randomly changed on a daily basis to simulate natural fluctuations in irradiance. Key properties including the photoperiod, minimum and maximum irradiance, daily average irradiance, and the overall shape of the light pattern were kept the same. Dynamic irradiance was provided by GreenPower LED top lighting compact modules (Philips Lighting).
Plants used to measure photosynthesis under natural fluctuating irradiance. The plants were brought to the greenhouse compartment to grow for another four weeks. Plants were grown on growth tables in the compartment of a Venlo-type glasshouse. One layer of cloth was put on the growth table and the greenhouse compartment had a photoperiod of 16 hours to allow for ample root growth. Only when global radiation outside the greenhouse dropped below 150 Wm −2 , were high-pressure sodium (HPS) lamps (600 W, Philips) used during the light period. These were switched off when outside global radiation increased to values above 250 Wm −2 . When the HPS lamps were on, the light intensity from these was approximately 150 μmolm −2 s −1 at plant level. The shading screen (HARMONY 4215 O FR, Ludvig Svensson) was closed when outside global radiation increased to values above 600 Wm −2 and was opened when outside global radiation dropped below 500 Wm −2 . Set points of day and night temperature were 22˚C and 18˚C, respectively. Relative humidity was set at 65% and plants were irrigated four times per day with tomato nutrient solution (see S1 File).

Measurements conducted
Unless stated otherwise, gas exchange measurements were conducted on the fourth or fifth leaf of four-week-old plants (after transplanting) using a portable gas exchange system (LI-6400XT, Li-Cor Bioscience) equipped with a 6 cm 2 leaf chamber fluorometer. Airflow was set to 500 μmols −1 during measurements and relative humidity was controlled at 75%. Irradiance was provided by a mixture of red (90%) and blue (10%) LEDs in the fluorometer.
Measurements to estimate V cmax and R d . Leaf temperature was kept around 25˚C. CO 2 response curves of photosynthesis (A n /c i curves) were measured by changing the atmospheric CO 2 concentration in the following order: 400, 300, 200, 100, 50, 400, 400, 500, 600, 800, 1000, 1200 ppm while keeping the light intensity at 1800 μmolm −2 s −1 . Each CO 2 concentration step took about 2-5 minutes to finish. Measured photosynthetic rates during A n /c i curve constructions were corrected for diffusion leaks according to the Li-Cor manual [26]. In total, eight replicates were obtained (see S1 File for details).
Measurements to parameterise function J(I(t)). Measurements were performed at an air temperature of 23˚C, and at two atmospheric CO 2 concentrations: 400 ppm and 800 ppm. For each CO 2 concentration, the leaf was exposed to a respective irradiance of 0, 50, 100, 200, 400, 600, 800, 1000 and 1200 μmolm −2 s −1 for at least 45 minutes to allow both leaf net photosynthetic rate and stomatal conductance to reach steady-state. Six replicates were obtained for each CO 2 concentration (refer to S1 File for details).

Measurements to track photosynthesis under natural fluctuating irradiance. Dynamic photosynthesis measurements were conducted between 3 and 24 September 2021 in a compartment (8×8 m) of a Venlo-type glasshouse located in Wageningen.
Measurements were conducted on leaves at the top of the plant that were fully exposed to sunlight and an air temperature of 23˚C. Two sets of atmospheric CO 2 concentrations, i.e. 400 ppm and 800 ppm were used. The photosynthetically active radiation (PAR) in the leaf chamber fluorometer was continuously adjusted to match the readings from an adjacent PAR sensor placed next to the chamber.
Gas exchange data were logged every two seconds between approximately 9:00 to 16:00 hours. In total, five measured replicates were obtained at 400 ppm CO 2 and one measurement was taken at 800 ppm CO 2 .

Parameter estimation
Values for the 3 unknown system parameters, k u , k d and c 3 , were estimated using Matlab's global optimisation function, the genetic algorithm (ga). This method is well suited to the

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optimisation of highly nonlinear problems and problems with a discontinuous objective function [27]. The parameters were inferred by minimising the objective function, Oðk u ; k d ; c 3 Þ � ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi P ð0:2ðg tc À g tc;m Þ 2 þ 0:2ðc i À c i;m Þ 2 þ 0:6ðA n À A n;m Þ 2 Þ N s ð19Þ where g tc,m , c i,m and A n,m denote the measured outputs defined in 13 and N is the number of observed data points, recorded every 2 seconds. By attributing weights to the individual terms, we ensured that an accurate prediction of the important metric A n was favoured. Measured g tc and A n values are shown in blue in Fig 5a and 5b. Initial conditions. Given that both g tc and c i are measured outputs, the initial conditions of the 2 state equations are known. Notice that the derived time constants k u = 179.4 s and k d = 830.3 s suggest that for a leaf under natural fluctuating irradiance, the overall stomatal dynamics related to an increase in irradiance is faster than those associated with a decrease in irradiance.

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The optimised c 3 value 37.96 (see Fig 4) increases the value of W c (defined in expression 4) compared to this function's value computed using the commonly used c 3 = 38.28 [28]. All parameter values in 15 and 16 are given in Table 2. Parameter values in 18 given in Table 3. Table 2. Values of the model parameters in expressions 15 and 16. Parameters of the steady-state target function G(I (t),c a (t)), computed under c a = 400ppm and c a = 800ppm, respectively, are also included. Estimated unknown system parameters are given with accompanying confidence intervals.

Parameter
Description Value  Table 3. Values of the model parameters in expression 18. A priori estimated parameters of function J(I(t)) and the respective steady-state parameters V cmax and R d are given. The unknown system parameter and its confidence interval is also shown. Finally, the parameters related to the temperature dependence of key FvCB model parameters are also given.

Photosynthesis under naturally fluctuating irradiance: Parameter estimation
We parameterise the model using data measured on 8 September 2021 under natural fluctuating light as this contains rich information pertaining to fluctuations between W c and W j limitation (refer to Fig 5c). Predictions for g tc are shown in red in Fig 5a. A 9% increase in predicted accuracy (from the objective function in 19) was obtained by modelling g tc with an asymmetric as opposed to a symmetric response to irradiance (see S1 File).
The dynamic relationship between the respective W c and W j limitations is shown in red in Fig 5c. Results indicate that Rubisco limited photosynthesis (W c ) coincides with elevated levels of natural irradiance seen Fig 1a. The model predicts that photosynthesis is limited by the electron transport rate (W j ) for the majority of the day, thus for irradiance levels that remain below 400 μmolm −2 s −1 . The correlation between A n and the rate of electron transport J(I(t)) is shown in Fig 6.

Photosynthesis under natural fluctuating irradiance: Model validation
We proceed by using the parameter values obtained in Tables 2 and 3 during model validation. The outcome is summarised in Table 4.
The results obtained for measurements taken on 6 Sept 2021 under an ambient CO 2 concentration of 400ppm are shown in Fig 7, whilst the results for measurements conducted on 7 Sept 2021 under a CO 2 concentration of 800ppm are shown in Fig 8. Both Figs 7b and 8b show good agreement between measured and modelled A n . Simulations shown in Fig 8c suggest that when c a is 800ppm, photosynthesis is solely limited by the electron transport rate. Furthermore, one observes similarities between the shapes of the modelled W j and A n . Given that W jc is computed using J(I(t)) from expression 11, this suggests that for tomato under the conditions reported here, it may be sufficient to use electron transport rates, inferred from steadystate data, in a dynamic setting. The correlation between A n and J(I(t)) is shown in Fig 9.

Discussion
It is highly unlikely that A n is in steady-state under natural fluctuating irradiance conditions and so observing natural dynamic as opposed to step-change responses to light is useful in aiding our understanding of this key photosynthetic property. However, nonsteady-state photosynthesis is often overlooked, with kinetic measurements of A n reported less due to the complexity associated with measuring and analysing them [29].

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We set out to develop a fit-for-purpose dynamic photosynthesis model. The model is both calibrated and validated using measurements taken under naturally fluctuating greenhouse conditions. Sufficiently accurate A n predictions in Table 4 suggest that the model (given in expressions [15][16][17] can potentially be used in greenhouse lighting control applications.

Model
Our model comprises 2 ODEs, predicting the total stomatal conductance to CO 2 (g tc ) diffusion and the CO 2 concentration inside a leaf (c i ). These predictions are required to compute the net photosynthetic rate (A n ) using Fick's law of diffusion (expression 14). Our results show that satisfactory fit for purpose A n values can be obtained by merely predicting the elements that comprise Fick's law of diffusion.
The dynamic binding of CO 2 inside a leaf is modelled using the FvCB model that has been adapted to predict the steady-state J values at different light intensities. Here, we used this

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application in a dynamic setting, defining the dynamic electron transport rate function, J(I(t)). It does not include the process of how J gradually increases after light increase.
Given that parameter V cmax is estimated from a priori A/c i measurements (see S1 File for details), we chose to optimise a parameter used to describe the temperature dependence of the Michaelis-Menten constant for CO 2 , c 3 , to increase the accuracy of our A n predictions (see Table 4).
The model is unique given its small size and the fact that it only comprises 3 unknown system parameters. We opt not to predict any detailed molecule concentrations such as RuBP (see [20] for example), and by assuming that the leaf is homogeneous, we do not include an additional equation that accounts for mesophyllic conductance (see [12] for example).
Furthermore, for V cmax as defined in the FvCB model [8], we too assume that Rubisco is activated [22]. Predictions for A n reported for tomato suggest that adequate A n values can be obtained under this stringent model assumption.

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Finally, it is important to realise that the implementation of a system which contains the FvCB model necessitates the a priori calculation of a multitude of species specific steady-state parameters. This requires repeated experimental measurements and some background knowledge on how to interpret and assimilate data.

Results
The first interesting point that emerges from using rapid fluctuating measurements to calibrate a model is the inferred time constants pertaining to the total stomatal conductance to CO 2 diffusion. Notice from Table 2 that k u , associated with an increase in irradiance is faster than k d . When inferring these parameter values from data measured after a single step change in irradiance, k u is slower than k d .
We highlighted the correlation between A n and the electron transport rate (see Figs 6 and 9). We know that the formation of ATP and NADPH molecules are dependent on the rate of electron transport and that this is light dependent. Accordingly, we observe a strong linear relationship between between A n and J under low irradiance levels. This indicates that under such conditions, photosynthesis is W j limited. Accurate results shown in Figs 7 and 8b in particular, suggest that: 1) it is sufficient to use an electron transport rate function, at least for our greenhouse-grown tomato leaves, calibrated using steady-state values at different irradiance levels, in a dynamic setting, and 2) the calibration of this function i.e. the parameter values J max , θ and γ, is critically important.

Conclusions
The importance of monitoring nonsteady-state responses to natural fluctuations in irradiance for improving crop photosynthesis has gained substantial support in recent years [30]. Our aim was to develop a small fit-for-purpose dynamic photosynthesis model that can be used in supplemental lighting control applications in greenhouses. We set out to build a model that accurately predicts the net photosynthetic rate (A n ) by taking plant physiology into account, and both calibrated and validated our model using nonsteady-state data measured under rapid fluctuating light conditions.
Four main points have emerged from our analysis: 1. We corroborated the added value of accounting for differences in stomatal responses to both increasing and decreasing fluctuations in irradiance. We observed a 9% increase in model accuracy when using 2 different time constants to describe the total stomatal conductance to CO 2 (refer to S1 File). In contrast, we observed no significant increase in the predicted accuracy of A n when modelling the steady-state target function of the total stomatal conductance to CO 2 , denoted by G, as a function of both irradiance and ambient CO 2 concentration (refer to S1 File).
2. We showed that incorporating the FvCB equations into a dynamic model is sufficient for obtaining accurate fit for purpose A n predictions under rapid natural light fluctuations. In particular, we found that the a priori parameterisation of the steady-state electron transport rate with respect to different irradiance levels (J(I(t)) is very effective in capturing the dynamics of photosynthesis in tomato leaves.
3. We showed the added value of optimising a parameter in one of the Michaelis-Menten constants in the FvCB model. Here, we opted to adjust parameter c 3 related to K n . Its value is often used from literature, despite being calibrated for different plant species. This cautions us when we are simply using parameter values from literature derived for different plant species. Accordingly, our model can predict dynamic photosynthesis for other crops, provided that parameters are adjusted, a priori, using both steady-state and dynamic measurements. 4. We showed that satisfactory photosynthesis results can be obtained even when a model does not account for complex biological factors such as enzymatic inhibition, liquid-gas interactions or chloroplast movement.